This invention relates to a method for simultaneously controlling the mean eccentricity and the longitudinal (East/West) motion of a geostationary satellite.
A method for controlling the East/West motion of geostationary satellites is of interest due to the stringent constraints imposed upon a satellite's longitudinal motion by ground-based antennas. The two methods of orbital control are referred to as North/South stationkeeping, which controls latitude excursions, and East/West stationkeeping, which controls longitude excursions. Though it is understood that there can be a cross-coupling effect during North/South stationkeeping maneuvers, attention herein is directed to the East/West control problem and neglects the effect of North/South maneuvers in the East/West direction.
With the increase of capability of the on-board payloads on satellites, there is a proportional increase in the power requirement of the payloads. The conventional solution is to enlarge the surface area of solar panels attached to the satellite in order to increase reception of available solar power. However, increasing the surface area of solar arrays enhances the perturbing force due to solar radiation pressure. Specifically, the solar panels act as sails driven by the solar wind with a force directly proportional to the area-to-mass ratio of the satellite.
Perturbations caused by solar radiation pressure result in the change in the magnitude of orbital effects of solar radiation pressure cause eccentricity magnitude to vary with a yearly period during which time the eccentricity magnitude will grow to a value that may cause apparent longitudinal motion (East/West motion) to exceed the limits of the permissible deadband. Eccentricity appears as daily longitudinal oscillations about the nominal location of the geostationary satellite as observed from a position on the Earth.
For geostationary satellites, the mean longitudinal acceleration is governed by the non-spherical effects of the Earth and the gravitational effects of the Sun and the Moon. In general, the Earth's tesseral harmonics are the dominant contribution to the longitudinal behavior, and their effects can be considered constant over longitudinal variations of less than 1 degree (&lt;1.degree.). However, for satellites near the Earth's gravitational equilibrium points, the longitudinal acceleration due to the Earth becomes small in comparison to that due to the Sun and the Moon. It is of interest that there are two stable equilibrium locations and two unstable equilibrium locations for satellites at geostationary altitude. The longitudinal acceleration caused by luni-solar effects dominate the mean longitudinal behavior near the equilibrium points and have significant effects away from the equilibrium points (FIG. 1). Therefore, an accurate model of the Sun's and Moon's effects on the orbital motion of the satellite must be used in order to target the longitude and longitudinal drift rate needed to initiate a proper free-drift cycle for any arbitrary satellite station location. It is necessary to account for a longitudinal acceleration that is not constant, but which is time varying in magnitude and, for station locations near an equilibrium point, in direction as well.
In order to appreciate the variability of longitudinal acceleration, reference is made to FIG. 1 which illustrates the long-periodic behavior of the longitudinal acceleration over a period of one month for various station longitudes of an object in geostationary orbit. The curves represent longitudinal accelerations in degrees per days squared (.degree./day.sup.2). The longitudinal acceleration is time-varying with a period of approximately 13.6 days. This graph illustrates the complexity of East/West stationkeeping, especially near an equilibrium point in that at the longitudinal acceleration can change direction. Moreover, due to variations in the gravity at different positions on the Earth, the value of the longitudinal acceleration will vary with the Earth's longitude.
An additional factor to consider is the solar radiation pressure effects on a spacecraft. Effects due to solar radiation pressure will perturb an orbit in such a way as to move the eccentricity vector perpendicular to the Earth-Sun line and in the direction of the orbital motion. If allowed to proceed naturally, this would result in an eccentricity vector roughly describing an ellipse with a peak magnitude possible greater than the allowable deadband and with a period of approximately one year. Superimposing the luni-solar effects onto this behavior serves to distort the ellipse and to cause local variations in the magnitude and rotation rate of the eccentricity vector. Depending upon the area-to-mass ratio of the satellite, the daily excursions of longitude could exceed the deadband by a significant factor.
The eccentricity vector extends from the Earth to the perigee. The value e.sub.y is equal to [e sin(.omega.)], and the value e.sub.x is equal to [e cos(.omega.)], where .omega. is the argument of perigee. FIG. 2 is a graph illustrating the mean eccentricity motion e.sub.x versus e.sub.y over a one-year period t.sub.0 to t.sub.T due to luni-solar gravitational effects and solar radiation pressure. The solar radiation pressure causes the annual elliptical motion, and the solar gravitation effects result in an ellipse which does not close over a period of one year. The smaller oscillations are primarily due to the Moon's gravitational effects on the satellite and have a period of approximately 13.6 days.
In order to meet the constraints established for controlling East/West stationkeeping of the satellite, the mean values of longitude and drift rate must be maintained to prevent any secular or long-periodic trends caused by the Earth's and the luni-solar gravitational effects from allowing the daily longitude variations to exceed the deadbands for a specified drift period between maneuvers. Furthermore, the mean eccentricity vector (which is manifested as the daily longitude variation) must be maintained below a preselected magnitude so that when superimposed upon the mean longitude, the net, or osculating, longitude will not exceed the specified deadband for the same drift period. Both of these conditions must be met simultaneously using the same set of maneuvers while making efficient use of propellant.
A search of prior art literature uncovered a number of references, only one of which appears to address the effects of the Sun and Moon on eccentricity targeting, although a few address the luni-solar effects on longitude and drift-rate targeting. The following is a summary of the references uncovered and their relevance.
M. C. Eckstein, "Station Keeping Strategy Test, Design and Optimization by Computer Simulation," Space Dynamics for Geostationary Satellites, Oct. 1985, Toulouse, France (CEPAD): This paper discusses stationkeeping with small longitudinal deadbands. The luni-solar effects are included by removing some of the deadband, that is, the effects due to the Sun and Moon are considered small, and a certain amount of the deadband is budgeted to these effects. This paper includes a study on the result of a simulation that simultaneously targets eccentricity and longitude to initiate a free-drift cycle and uses the Sun-Pointing Perigee Strategy (SPPS) to target eccentricity. (SPPS is a method for controlling the eccentricity vector by maintaining it in the general direction of the Sun.) The targeting scheme used therein excludes the Sun's and the Moon's gravitational effects on eccentricity. This paper does not address the possibility of a variable longitudinal acceleration due to luni-solar effects, and it assumes that acceleration will be constant in direction and magnitude. This results in a need for an iterative control scheme requiring some operator intervention. This problem is addressed by the present invention.
A. Kamel and C. Wagner, "On the Orbital Eccentricity Control of Synchronous Satellites," The Journal of the Astronautical Sciences, Vol. 30, No. 1 (Jan.-Mar. 1982), pp. 61-73: This paper contains many of the fundamental ideas behind the current strategy. The eccentricity targeting strategy therein used the Sun Pointing Perigee Strategy (SPPS) but only included the effects of the solar radiation pressure on eccentricity, thus excluding the luni-solar effects. The calculation of the longitudinal acceleration took into account only the Earth's gravitational effects (tesseral harmonics). The paper also suggests allotment of the luni-solar effects on eccentricity into the deadband budget, thereby tightening the usable deadband and costing more propellant. The present work represents an advancement over this simplification, particularly where the limited deadband constraints renders it unfavorable to include the luni-solar effects by removing part of the available deadband.
C. F. Gartrell, "Simultaneous Eccentricity and Drift Rate Control," Journal of Guidance and Control, Vol. 4, No. 3 (May-Jun. 1981), pp. 310-315: This paper describes a method for simultaneous control of eccentricity and drift rate using equations which only take into account solar pressure on orbit eccentricity. The spacecraft considered in this analysis is quite small, so the effect of solar radiation pressure on eccentricity is not large. Thus, the maneuvers required to control drift rate are not required to control eccentricity magnitude, therefore, only one maneuver is required. The maneuver strategy is not generalized to N-maneuvers (where N is an arbitrary number of maneuvers), and the equations do not consider the gravitational effects of the Sun and the Moon in the targeting strategy.
E. M. Soop, Introduction to Geostationary Orbits, European Space Agency (Nov. 1983), Publication SP-1053, pp. 1-143: This work provides a fairly broad explanation of orbital control of geostationary satellites. Particular attention is directed to Chapter 7 relating to longitude stationkeeping. It discusses the Sun-Pointing Perigee Strategy to target eccentricity and the use of two maneuvers to simultaneously target both drift rate and eccentricity in order to maintain the longitudinal deadband.
A. Kamel et al., "East-West Stationkeeping Requirements of Nearly Synchronous Satellites Due to Earth's Triaxiality and Luni-Solar Effects," Celestial Mechanics, Vol. 8 (1973), (D. Reidel Publishing Co., Dordrecht-Holland), pp. 129-148: This paper describes the equations for the disturbing function (which contains Earth's gravitational effects, as well as the luni-solar gravitational effects). This papers indicates that by not including the perturbations in the orbit caused by the Moon's and Sun's gravitational effects, longitude and drift rate will be incorrectly targeted. Eccentricity targeting is not mentioned in this paper, nor was any strategy suggested here which would achieve the proper target drift and longitude. Kamel has written several papers which contain a description of an analytic solution addressing the geostationary stationkeeping problem. Not all of those are discussed herein.
D. J. Gantous, "Eccentricity Control Strategy for Geosynchronous Communications Satellites," Telesat Canada, May 1987: This paper describes an amended Sun-Pointing Perigee Strategy based on earlier work of Kamel and Wagner. The author observes that the natural motion of the eccentricity vector caused by the solar radiation pressure and the luni-solar gravitational effects will have a linear trend, i.e., the eccentricity plot does not achieve a closed ellipse after a year (as can be seen in FIG. 2). The targeting strategy described therein accounts for this phenomenon. However, the targeting strategy does not consider the local variations due to the gravitational effects of the Sun and the Moon.
J. A. Kechichian, "A Split .DELTA.V for Drift Control of Geosynchronous Spacecraft," AIAA/AAS 21st Aerospace Sciences Meeting, (Jan. 10-13, 1983), AIAA-83-0017: This paper discusses the use of a two-maneuver technique over a 24-hour period in order to maintain a longitude deadband of .+-.0.05.degree. while simultaneously controlling longitude and eccentricity. The author discusses targeting osculating effects, whereas the present invention addresses targeting mean values. The author concludes that his technique will deviate from a specified deadband limit after only a few cycles.
S. Parvez et al., "East-West Stationkeeping Near an Equilibrium Longitude at 105 Degrees West," AAS/AIAA Astrodynamics Specialist Conference, AAS-87-545 (Aug. 10-13, 1987), pp. 1-14: This paper describes stationkeeping near a stable equilibrium point. This paper commented on the problems of predicting spacecraft longitudinal motion after a change in orbital velocity due to a required attitude adjustment. This paper did not consider the luni-solar effects and seemed mainly a plea for an explanation for the apparently high sensitivity of longitude motion to attitude maneuvers and the lack of a predictable drift cycle.
Various other papers were uncovered of less relevance than those mentioned above. Further, a search of the Patent Office records relating to this invention uncovered the following patents, none of which are particularly relevant to the problem at hand.
U.S. Pat. No. 4,767,084 to Chan et al. uses momentum unloads to control stationkeeping. There are no instructions on how to control the orbit, and this is a scheme that may not be appropriate for spacecraft with large area-to-mass ratios because inadequate thruster power is available.
U.S Pat. No. 4,521,855 to Lehner et al. relates to physically controlling the attitude of the satellite and does not relate to stationkeeping.
U.S. Pat. No. 4,599,697 to Chan et al. also relates to spacecraft attitude control.
U.S. Pat. No. 4,537,375 to Chan also relates to attitude control.
U.S. Pat. No. 4,776,540 to Westerlund relates to attitude control of a satellite to compensate for changes in latitude.
U.S. Pat. No. 4,837,699 to Smay et al. also relates to attitude control, but in the context of a spinning satellite.
In view of the foregoing, what is needed is a generalized method for East/West stationkeeping and targeting of geosynchronous satellites that uses an arbitrary number of maneuvers and which takes into account luni-solar effects on both eccentricity and longitudinal motion.